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Full text of "Analytical Mechanics"

I. BINOMIAL THEOREM.
(a + z)ğ = a" + £ a"-* z + W(n~1} ağ-v- + Ğ(n-lKn-2)
When a; -C a, and consequently '
/       T\            en a; -    a, an
= an  1+-).   U2  *3    ,     '
V       oj         '3'etC"<<a:-
Applying this theorem to (1 db x)-1 we obtain 1
= 1 - x + re2 -
1+0?
H i — x    P^*1611 x ^ 1> an(* consequently"! [jr2, re3, etc.3<£.                        J
-J— =   1  + x + X2 + XZ +   .   .   . 1  — X
["When x <C 1, and consequently!
'
II.   QUADRATIC FORMULA.
If x satisfies the quadratic equation axz + bx + c = 0, then
__ — 6 d= Vb2 — 4 ac X~           2^
III.   LOGARITHMIC RELATIONS. (a)     log ab = log a + log 6.
n N     ,      n       T          fThis formulse may be obtained from (a) by"]
(b)     loga" = nloga.   ^^ ft = ^ ^^ ^ a6 = ^       /J
(c)      log ^ = log a — log 6.   [This follows immediately from (a) and (b) .]
(d)     log 1=0.   [This is obtained by letting b = a in (c).]
IV. TRIGONOMETRIC RELATIONS.